An endomorphism on immersed curves in the pillowcase

TL;DR

This paper studies the holonomy-perturbed traceless SU(2) character variety of certain tangles, representing endomorphisms as immersed curves in the pillowcase, and shows they have identical images under a specific functor.

Contribution

It introduces a novel approach to represent endomorphisms of tangles as immersed curves in the pillowcase and analyzes their images in the Weinstein symplectic category.

Findings

Endomorphisms are expressed via doubling and figure eight operations.

The images of these endomorphisms in the pillowcase are shown to be the same.

The work connects tangle invariants with symplectic geometry through immersed curves.

Abstract

We examine the holonomy-perturbed traceless SU(2) character variety of the trivial four-stranded tangle {p_1,p_2,p_3,p_4} X [0,1] in S^2 X [0,1] equipped with a strong marking, either an earring or a bypass. Viewing these marked tangles as endomorphisms in the cobordism category from the four-punctured sphere to itself, we identify the images of these endomorphisms in the Weinstein symplectic partial category under the partially defined holonomy-perturbed traceless character variety functor. We express these endomorphisms on immersed curves in the pillowcase in terms of doubling and figure eight operations and prove they have the same image.